# complementary subspace

## Direct sum decomposition.

Let $U$ be a vector space, and $V,W\subset U$ subspaces. We say that $V$ and $W$ span $U$, and write

 $U=V+W$

if every $u\in U$ can be expressed as a sum

 $u=v+w$

for some $v\in V$ and $w\in W$.

If in addition, such a decomposition is unique for all $u\in U$, or equivalently if

 $V\cap W=\{0\},$

then we say that $V$ and $W$ form a direct sum decomposition of $U$ and write

 $U=V\oplus W.$

In such circumstances, we also say that $V$ and $W$ are complementary subspaces, and also say that $W$ is an algebraic complement of $V$.

Here is useful characterization of complementary subspaces if $U$ is finite-dimensional.

###### Proposition 1

Let $U,V,W$ be as above, and suppose that $U$ is finite-dimensional. The subspaces $V$ and $W$ are complementary if and only if for every basis $v_{1},\ldots,v_{m}$ of $V$ and every basis $w_{1},\ldots,w_{n}$ of $W$, the combined list

 $v_{1},\ldots,v_{m},w_{1},\ldots,w_{n}$

is a basis of $U$.

Remarks.

• Since every linearly independent subset of a vector space can be extended to a basis, every subspace has a complement, and the complement is necessarily unique.

• Also, direct sum decompositions of a vector space $U$ are in a one-to correspondence fashion with projections on $U$.

## Orthogonal decomposition.

Specializing somewhat, suppose that the ground field $\mathbb{K}$ is either the real or complex numbers, and that $U$ is either an inner product space or a unitary space, i.e. $U$ comes equipped with a positive-definite inner product

 $\langle,\rangle:U\times U\rightarrow\mathbb{K}.$

In such circumstances, for every subspace $V\subset U$ we define the orthogonal complement of $V$, denoted by $V^{\perp}$ to be the subspace

 $V^{\perp}=\{u\in U:\langle v,u\rangle=0,\text{ for all }v\in V\}.$
###### Proposition 2

Suppose that $U$ is finite-dimensional and $V\subset U$ a subspace. Then, $V$ and its orthogonal complement $V^{\perp}$ determine a direct sum decomposition of $U$.

Note: the Proposition is false if either the finite-dimensionality or the positive-definiteness assumptions are violated.

Title complementary subspace ComplementarySubspace 2013-03-22 12:52:16 2013-03-22 12:52:16 rmilson (146) rmilson (146) 11 rmilson (146) Definition msc 15A03 algebraic complement complementary direct sum decomposition orthogonal complement